ReLU Barrier Functions for Nonlinear Systems with Constrained Control: A Union of Invariant Sets Approach
This addresses safety certification for nonlinear control systems with constraints, which is an incremental improvement over existing barrier function methods.
The paper tackles the problem of certifying safety for nonlinear systems with input constraints by proposing an approximation-verification pipeline using piecewise-affine surrogates and a Union of Invariant Sets approach, resulting in larger certified invariant sets than linear designs while maintaining tractable computation time in pendulum and cart-pole simulations.
Certifying safety for nonlinear systems with polytopic input constraints is challenging because CBF synthesis must ensure control admissibility under saturation. We propose an approximation--verification pipeline that performs convex barrier synthesis on piecewise-affine (PWA) surrogates and certifies safety for the original nonlinear system via facet-wise verification. To reduce conservatism while preserving tractability, we use a two-slope Leaky ReLU surrogate for the extended class-$\mathcal{K}$ function $α(\cdot)$ and combine multiple certificates using a Union of Invariant Sets (UIS). Counterexamples are handled through local uncertainty updates. Simulations on pendulum and cart-pole systems with input saturation show larger certified invariant sets than linear-$α$ designs with tractable computation time.